Quincke rotation, a well-observed phenomenon in particle suspensions, denotes the spontaneous rotation of dielectric particles immersed in a slightly dielectric liquid when subjected to a high enough DC electric field. Quincke rotation occurs when the charge relaxation time of the particles is greater than that of the fluid medium, causing the particles to become polarized in a direction opposite to that of the electric field and therefore giving rise to an unstable equilibrium position. When slightly perturbed, the particles start to rotate, and if the applied electric field exceeds a critical value this perturbation does not decay and the particle rotation reaches a steady state with a constant angular velocity obtained by balancing the viscous torque with the electric torque due to the induced dipole. The dynamics of a particle undergoing Quincke rotation have been previously shown to obey the classic Lorenz oscillator equations with two bifurcations. When the applied electric field exceeds the critical electric field for spontaneous rotation, the angular velocity undergoes a supercritical pitchfork bifurcation, by which the zero angular velocity state becomes unstable and a stable state is reached where the angular velocity depends on the ratio of the applied electric field to the critical field. Upon increasing the electric field further, the angular velocity undergoes a subcritical Hopf bifurcation and becomes erratic indicating Lorenz chaos.
A useful application of Quincke rotation lies in its ability to modify the effective rheological properties of suspensions under flow. When a suspension undergoing Quincke rotation is subjected to a steady shear flow, its apparent viscosity has been shown to decrease as a result of the enhanced rotation rate of the particles with respect to the flow vorticity, which has the effect of increasing the flow rate and therefore decreasing the suspension viscosity. An apparent increase in the effective conductivity of the suspension has also been reported. Most previous studies of these two effects have focused on the dynamics of a single isolated spherical particle and therefore are unable to capture interactions between particles in semi-dilute or concentrated suspensions. In particular, experiments in suspensions typically exhibit weaker angular velocities than predicted by the theory for an isolated sphere. Also, the critical field above which rotation takes place has been found to be slightly higher than that predicted by the single sphere theory.
In this work, we use a combination of numerical simulations and asymptotic theory to study the effect of electrohydrodynamic interactions between particles on Quincke rotation. We study the prototypical case of two equally sized spheres carrying no net charge and interacting with each other both electrically and hydrodynamically. We use the classic method of reflections to capture far-field interactions, and solve a coupled system of time-dependent ordinary differential equations for the dipole moments, angular velocities, and positions of the two spheres capturing interactions up to order O$(R^{-3})$ (where \textit{R} is the separation distance between the two sphere centres).
We first perform numerical simulations of this coupled system. In the case when the spheres are held in place, we find that Quincke rotation occurs in the presence of interactions and the magnitude of the angular velocity is weaker than in the isolated case. When the particles are free to move, dipolar forces between particles and the rotlet flow driven by the rotation of the spheres also results in translational motions. In particular, dipolar interactions result in attraction in the direction of the applied field, but in repulsion in transverse direction. In both cases, we find that the critical field above which rotation takes place is more than that predicted by the single sphere theory, which is consistent with experimental observations. In the case of freely suspended spheres, we also model the dynamics of the spheres upon contact by assuming that no slip occurs between the two touching surfaces. We find that the two spheres come into contact, and subsequently remain in contact. This suggests that clustering likely happens in rheological experiments, and multiparticle simulations may be required to investigate the structures developing in these suspensions. We also perform a stability analysis on the base state of no-rotation for a pair of spherical particles. This helps us in determining the critical electric field for a given configuration of the two spherical particles.
Finally, the steady-state velocity that the two spheres reach in the case is also obtained using the method of reflections as an asymptotic expansion, where we find that the leading-order correction to the steady angular velocity due to interactions is of order O$(R^{-3})$.